SGTSVX(3)

Hey bestie! 💖 This is SGTSVX(3), freshly unfolded from the Library Functions Manual. The page tag on this one is /home/abui...RC/sgtsvx.f(3), and it's giving tridiagonal linear solver main-character energy. ✨

🌸 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sgtsvx.f

That's the source file this page is serving, bestie. 💅

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✨ SYNOPSIS

🧩 Functions/Subroutines

• subroutine SGTSVX — SGTSVX computes the solution to system of linear equations A * X = B for GT matrices.

subroutine SGTSVX (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2,
    ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)

📖 Function/Subroutine Documentation

Here's the full typed signature in all its glory — every type, every dimension, every argument, no skips:

subroutine SGTSVX (character fact, character trans, integer n, integer nrhs,
    real, dimension( * ) dl, real, dimension( * ) d, real, dimension( * )
    du, real, dimension( * ) dlf, real, dimension( * ) df, real, dimension(
    * ) duf, real, dimension( * ) du2, integer, dimension( * ) ipiv, real,
    dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer
    ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr,
    real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SGTSVX computes the solution to system of linear equations A * X = B for GT matrices. 🫶

🎯 Purpose

Okay so SGTSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B or A**T * X = B, where A is a tridiagonal matrix of order N and X and B are N-by-NRHS matrices.

It also hands you error bounds on the solution and a condition estimate. Obsessed. ✨

🧠 Description

The routine does these steps, in this exact order:

1. If FACT = 'N', the LU decomposition is used to factor the matrix A as A = L * U, where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
2. If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
3. The system of equations is solved for X using the factored form of A.
4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Bestie, step 2 is the dramatic one: if U(i,i)=0, that factor is exactly singular and the routine bails with INFO = i. 💀

🎀 Parameters

• FACT — CHARACTER*1. Specifies whether or not the factored form of A has been supplied on entry.
  • = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not be modified.
  • = 'N': The matrix will be copied to DLF, DF, and DUF and factored.
• TRANS — CHARACTER*1. Specifies the form of the system of equations:
  • = 'N': A * X = B (No transpose)
  • = 'T': A**T * X = B (Transpose)
  • = 'C': A**H * X = B (Conjugate transpose = Transpose)
• N — INTEGER. The order of the matrix A. N >= 0.
• NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• DL — REAL array, dimension (N-1). The (n-1) subdiagonal elements of A.
• D — REAL array, dimension (N). The n diagonal elements of A.
• DU — REAL array, dimension (N-1). The (n-1) superdiagonal elements of A.
• DLF — REAL array, dimension (N-1).
  • If FACT = 'F', then DLF is an input argument and on entry contains the (n-1) multipliers that define the matrix L from the LU factorization of A as computed by SGTTRF.
  • If FACT = 'N', then DLF is an output argument and on exit contains the (n-1) multipliers that define the matrix L from the LU factorization of A.
• DF — REAL array, dimension (N).
  • If FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
  • If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
• DUF — REAL array, dimension (N-1).
  • If FACT = 'F', then DUF is an input argument and on entry contains the (n-1) elements of the first superdiagonal of U.
  • If FACT = 'N', then DUF is an output argument and on exit contains the (n-1) elements of the first superdiagonal of U.
• DU2 — REAL array, dimension (N-2).
  • If FACT = 'F', then DU2 is an input argument and on entry contains the (n-2) elements of the second superdiagonal of U.
  • If FACT = 'N', then DU2 is an output argument and on exit contains the (n-2) elements of the second superdiagonal of U.
• IPIV — INTEGER array, dimension (N).
  • If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the LU factorization of A as computed by SGTTRF.
  • If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the LU factorization of A; row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required.
• B — REAL array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
• LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• X — REAL array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
• LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N).
• RCOND — REAL. The estimate of the reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.
• FERR — REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
• BERR — REAL array, dimension (NRHS). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
• WORK — REAL array, dimension (3*N).
• IWORK — INTEGER array, dimension (N).
• INFO — INTEGER.
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, and i is
    • <= N: U(i,i) is exactly zero. The factorization has not been completed unless i = N, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
    • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

👑 Author

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Definition at line 290 of file sgtsvx.f.

🪄 Author — auto-generated edition

Generated automatically by Doxygen for LAPACK from the source code.

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Final footer vibes, bestie: LAPACK Version 3.12.0 and the page tag /home/abui...RC/sgtsvx.f(3). Go off and solve those tridiagonal systems. 💖✨